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Two Rolling Balls (Posted on 2009-11-04) Difficulty: 3 of 5
Two identical balls roll across the top of a table on parallel paths. One of the balls has to roll down into and up out of a dip in the table. The other ball rolls on the flat all the time. Which ball gets to the far side of the table first, and why?.

No Solution Yet Submitted by Brian Smith    
Rating: 3.5000 (4 votes)

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The dip | Comment 4 of 21 |
The text says that the we have "the top of a table" which is implying that the surface is horizontal.

The shape of the "dip" does seem paramount. 

I'm adding a few assumptions.  Firstly no resistance or force of any kind, other than gravity, is to be considered.  Secondly the transition, both in and out of the dip, is a "smooth" curve, no 'sharp' edge like that of a gutter in a pavement.

When entering the dip, the ball has two vector components in its motion, a constant  horizontal one which matches that of the ball on the flat, and a vertical one which accelerates under gravity as it descends and decelerates as it rises (the deceleration cancels the effect of the acceleration).

That therefore means that both balls will arrive at the same time.

Again, re the shape of the dip.
Above I was assuming a dip with a profile which is mirrored on both sides of a vertical axis.  That need not be, just let the profile be smooth so as the ball always remains in contact with the surface and is not impeded such that it might have cause to jump.

I am vaguely recalling something from my Physics of so many years distant about projectiles where the horizontal component of its motion is constant even though the vertical is influenced by gravity.  In an 'ideal' situation this is the same, yes?

Edited on November 5, 2009, 9:04 am
  Posted by brianjn on 2009-11-04 19:47:38

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